On the Stokes-type resolvent problem associated with time-periodic flow around a rotating obstacle

TL;DR

This paper develops a new Sobolev space framework to analyze the resolvent problem for viscous flow around a rotating obstacle, enabling solutions for all purely imaginary parameters and advancing understanding of time-periodic flows.

Contribution

Introduces a homogeneous Sobolev space approach to establish well-posedness of the resolvent problem on the imaginary axis for rotating flows.

Findings

Existence of unique solutions in homogeneous Sobolev spaces.

Derivation of uniform resolvent estimates.

Application to time-periodic linear flow problems.

Abstract

Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.